LINEAR DIFFERENTIAL EQUATIONS 



T.4BLE A 



359 



Example 3, Fig. 5 



y" 



v(l) 



-V + 



1, y(i) = 



\ < X< 00 



Exact solution: y{;x) = sin (x — 1) + - cos (x — 1) 



X 



(a) Wave perturbation, with the initial conditions satisfied exactly 



ITo = cos (x — 1) 



Wi = 2 sin (a: - 1) - 2 cos (x + 1) (Ci 2x - Ci 2) 

 - 2 sin {x + 1) (Si 2x - Si 2) 



(b) Wave perturbation, matching the exact solution at infinity 



Wo = sin (x - 1) 



Wi = 2 sin (x + 1) Ci 2x - 2 cos (x + l)(Si 2x - x/2) 



This is an example of a solution in an infinite interval, where the per- 

 turbation term is not small throughout. It is interesting to note that the 

 second form gives good agreement with the accurate solution in most of 

 the range of integration. 



